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This book presents an overview of the research in ultrametric summability theory. It will appeal to both young researchers and more experienced mathematicians who are looking to explore new areas in analysis.

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Ultrametric analysis has emerged as an important branch of mathematics in recent years. This book presents, for the first time, a brief survey of the research to date in ultrametric summability theory, which is a fusion of a classical branch of mathematics (summability theory) with a modern branch of analysis (ultrametric analysis). Several mathematicians have contributed to summability theory as well as functional analysis. The book will appeal to both young researchers and more experienced mathematicians who are looking to explore new areas in analysis.

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Preface.- Introduction and Preliminaries.- Some Arithmetic and Analysis in Qp : Derivatives in Ultrametric Analysis.- Ultrametric Functional Analysis.- Ultrametric Summability Theory.- References.- Index.

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Most of the material discussed in the book An Introduction to Ultrametric Summability Theory has not appeared in a book form before. The Silverman Toeplitz theorem in an ultra-metric field turns into the following form: an infinite matrix or a summability matrix A = ( a nk ) preserves convergent sequences if and only if sup n ; k a nk < Yen ; lim n © Yen a nk = d k exists, k = 0, 1, 2, and lim n © Yen exists. The theorem is proved by using the sliding-hump method avoiding traditional tools like the signum function. Schur s theorem, Steinhaus theorem, and Knopp s core theorem are discussed. It is proved that neither certain Steinhaus type theorems nor the Mazur Orlicz theorem and Brudno s theorem holds. Certain special summability methods, such as the Norlund method, weighted mean method, Euler method, Taylor method and their properties are extensively discussed. Some product theorems and Tauberian theorems are also proved. Double sequences and double series are discussed, and the Silverman Toeplitz theorem for four-dimensional infinite matrices is given. Some applications of this theorem are also discussed in the book. ( Hüseyin €akalli, Maltepe University, Istanbul, Turkey ) - See more at:

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